At the end of every relationship, there are always certain things that you're told you should do. Block them on social media, delete the texts and pictures, maybe give them that one sweatshirt back. If the relationship lasted long enough, you may have given or received gifts from this person, and once it's over, it's hard to decide what to do with them. Here's how I like to go through and choose what to do.
How the relationship ended is a good indicator of how to go through all of the stuff that you collected in the relationship. A mutual breakup is a bit easier to deal with; perhaps you both realized it just wasn't working and parted ways as friends, and so any tangible reminders of the relationship are simply objects with a fond memory attached. If you liked the gift, go ahead and keep it. If you only pretended to like it (we've all been there), re-gift it, donate it, or throw it away. If you and your ex are on good terms, it might be courteous to ask them if they would like anything back, maybe something that cost a lot or was extremely meaningful to them that you wouldn't want to keep.
A messy breakup is much more difficult to clean up after, especially if you're not the one who wanted it to end. Suddenly you have all these reminders of what you lost, and it absolutely sucks to look at them, but don't feel like you have to get rid of everything; there may be things that you really like and would later regret getting rid of. Find a safe place to keep them out of sight until you're ready to pull it out again. Then there are the things that have a lot more emotional weight to them that at the time you can't bear to get rid of. That's okay; take your time, wait until you're ready to let go.
Then there are the relationships that not only were a messy breakup, but your significant other cheated or lied, and now you're out for blood. First of all, murder is illegal, folks, so keep the revenge plots in check. Use what you have from that person to get rid of some of your anger. Love notes make great kindling; get together with your friends and have a night out under the stars around a campfire! Even though you might not want to ever speak to them again, if you have something very valuable to them, it'd be best to give it back to them. Sometimes you just have to suck it up and be the bigger, more civil person.
There are always things that you feel fine keeping from relationships, like maybe they got you a cool poster of a band you like, or socks with quotes from your favorite book on them. These are things that don't really have any connection to the relationship itself, so looking at them every day won't make you sad. I like to think of these as spoils of war; you got some cool stuff out of a less-than-great situation. Of course, everyone deals with breakup differently, so you do whatever works best for you, and good luck!